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A colony of $100$ bacteria doubles in size every $60$ hours. What will the population be $180$ hours from now $?$

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Exponential growth and decay: word problems

Full solution

Q. A colony of

100

bacteria doubles in size every

60

hours. What will the population be

180

hours from now

?

Identify Parameters: Identify the initial population, the growth rate, and the total time for growth. $\newline$ Initial population $P_0$ = $100$ bacteria $\newline$ Growth rate = doubles every $60$ hours $\newline$ Total time $t$ = $180$ hours $\newline$ Determine the number of doubling periods within the total time.
Calculate Doubling Periods: Calculate the number of doubling periods. $\newline$ Doubling period $T$ = $60$ hours $\newline$ Number of doubling periods $n$ = Total time / Doubling period $\newline$ $n = \frac{180 \, \text{hours}}{60 \, \text{hours}}$ $\newline$ $n = 3$
Apply Exponential Growth Formula: Apply the formula for exponential growth. $\newline$ Final population $P$ = Initial population $*$ (Growth factor)^Number of doubling periods $\newline$ Growth factor for doubling = $2$ $\newline$ $P = 100 * 2^n$ $\newline$ $P = 100 * 2^3$
Calculate Final Population: Calculate the final population. $\newline$ P = $100 \times 2^3$ $\newline$ P = $100 \times 8$ $\newline$ P = $800$

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